solve the following inequality algebraically. |x + 6| > 6

Explanation:

Step1: Split into two cases

When (x + 6\geq0) (i.e., (x\geq - 6)), the inequality (|x + 6|>6) becomes (x + 6>6). Solving for (x): (x+6>6\Rightarrow x>6 - 6\Rightarrow x>0). Since (x\geq - 6), the valid part of this solution is (x>0). When (x + 6<0) (i.e., (x<-6)), the inequality (|x + 6|>6) becomes (-(x + 6)>6).

Step2: Solve the second - case inequality

Expand (-(x + 6)>6) to (-x-6>6). Add 6 to both sides: (-x>6 + 6\Rightarrow -x>12). Multiply both sides by - 1 and reverse the inequality sign: (x<-12). Since (x<-6), the valid part of this solution is (x<-12).

Answer:

(x<-12) or (x > 0)